Network selection method and apparatus for integrated cellular and drone-cell networks

Abstract

Provided are a network selection method and an apparatus for integrated cellular and drone-cell networks, where the method includes: acquiring a dynamic network model and a dynamic user model, generating a random event vector according to the dynamic network model and the dynamic user model; generating an action vector according to the random event vector; obtaining an individual utility of each user according to the action vector and the random event vector; constructing a first selection model; obtaining the value of the action probability according the first selection model to determine networks that the users choose to access according to the value of the action probability.

Description

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to Chinese Patent Application No. 201811353219.4, filed on Nov. 14, 2018, which is hereby incorporated by reference in its entirety.

TECHNICAL FIELD

The present disclosure relates to the field of a network selection, and in particular to a network selection method and an apparatus for integrated cellular and drone-cell networks.

BACKGROUND

In order to provide better network dada service and alleviate the congestion of cellular network, a solution of resorting to drone-cells (i.e., low-altitude drones equipped with transceivers) to offload traffic from the congested cellular network can be adopted. To make full use of resources of drone-cells, a significant challenge is how to make an efficient and fair network selection for integrated cellular and drone-cell networks.

An approach for solving a network selection problem is a game-theory-based approach. In the game-theory-based approach, the network selection problem is firstly modeled as a game and then a centralized/distributed approach is exploited to achieve an equilibrium. For example, Cheung et al. formulated the network selection problem as a Bayesian game under the condition that mobility information of users is not complete. Then, they proposed a distributed approach with good convergence properties to achieve the Bayesian Nash equilibrium.

The existing game-theory-based approaches consider the interaction and competition among users. Most of the approaches, however, studied the network selection problem under the situation with quasi-static or predictable network states. However, the integrated cellular and drone-cell networks are highly dynamic, and the network state is hard to predict. Therefore, it is difficult for the existing game-theory-based approaches to solve the network selection problem of the integrated cellular and drone-cell networks.

SUMMARY

The present disclosure provides a network selection method and an apparatus for integrated cellular and drone-cell networks, aiming at solving the problem that the existing game-theory-based approaches cannot solve the network selection problem for the integrated cellular and drone-cell networks because the integrated cellular and drone-cell networks are highly dynamic, and the network state is hard to predict.

In a first aspect, the present disclosure provides a network selection method for integrated cellular and drone-cell networks, including:

acquiring a dynamic network model and a dynamic user model, where the dynamic network model includes at least a location model of the drone-cell networks, a capacity model of the cellular network and a capacity model of the drone-cell networks, and the dynamic user model includes at least a location model and a transmission rate model of users;

generating accessible network sets of the users according to the location model of the drone-cell networks and the location model of the users;

generating a random event vector according to the capacity model of the cellular network, the capacity model of the drone-cell networks, the accessible network sets of the users and the transmission rate model, where the accessible network sets of the users include the drone-cell networks and/or the cellular network:

generating an action vector according to the random event vector where the action vector is used for indicating that the users choose to access to the drone-cell networks and/or the cellular network:

obtaining an individual utility of each user according to the action vector and the random event vector;

constructing a first selection model, where the first selection model includes a first objective function and a first constraint, where the first objective function is a proportional fairness function with time average of individual utilities of the users as independent variables, and the first constraint includes at least a first coarse correlated equilibrium constraint, a first minimum individual time average utility constraint and a first action probability constraint, where the first coarse correlated equilibrium constraint is used for constraining the time average of the individual utilities and first auxiliary variables, the first minimum individual time average utility constraint is used for constraining the time average of the individual utilities, and the first action probability constraint is used for constraining the action probability under the condition of the random event vector;

obtaining the time average of the individual utilities according to the individual utilities, the random event probability and the action probability under the condition of the random event vector, where the action probability under the condition of the random event vector is the probability that the users execute the action vector under the condition that the random event vector occurs, and the random event probability is the probability that the random event occurs; and

obtaining a value of the action probability according the first selection model, and determining networks that the users choose to access according to the value of the action probability.

The network selection method provided by the present disclosure includes acquiring the dynamic network model and the dynamic user model, generating the random event vector according to the dynamic network model and the dynamic user model, constructing the first selection model according to the random event vector and the action vector, obtaining the value of the action probability according to the first selection model, and determining the networks that the users choose to access according to the value of the action probability. It solves the problem that the existing game-theory-based approaches cannot solve the network selection problem for the integrated cellular and drone-cell networks because the integrated cellular and drone-cell networks are highly dynamic, and the network state is hard to predict.

In a second aspect, the present disclosure provides a network selection apparatus for integrated cellular and drone-cell networks, including:

a transceiver, configured to collect capacity information of the drone-cell networks, capacity information of the cellular network, set information of accessible networks of the users, data transmission rate information, and transmitting action vector information to the users so that the users can determine the networks to access according to the action vector information;

a processor, configured to generate the action vector information according to the capacity information of the drone-cell networks, the capacity information of the cellular network, the set information of the accessible networks of the users, the data transmission rate information and a fourth selection model.

The fourth selection model is that a difference value between drift of total violation and an utility is less than or equal to a penalty upper bound, where

the drift of the total violation is obtained according to a value of the total violation at a current time slot and a value of the total violation at a next time slot;

the value of the total violation at the current time slot is obtained according to a first virtual value at the current time slot, a second virtual value at the current time slot and a third virtual value at the current time slot;

the first virtual value in the first virtual queue at the current time slot is generated according to violation of second coarse correlated equilibrium constraints at a previous time slot and the first virtual value in the first virtual queue at the previous time slot;

the second virtual value in the second virtual queue at the current time slot is generated according to violation of third auxiliary variable constraints at the previous time slot and the second virtual value in the second virtual queue at the previous time slot;

the third virtual value in the third virtual queue at the current time slot is generated according to violation of a second minimum individual time average utility constraint at the previous time slot and the third virtual value in the third virtual queue at the previous time slot, where the first virtual value at an initial time slot, the second virtual value at the initial time slot, and the third virtual value at the initial time slot are all zero;

a third selection model includes a third objective function and a third constraint, where the third objective function is time average expectation of a proportional fair function with third auxiliary variables as independent variables, and the third constraint includes at least the second coarse correlated equilibrium constraints, the second minimum individual time average utility constraint, a second auxiliary variable constraint, and the third auxiliary variable constraints, where the second coarse correlated equilibrium constraints are used for constraining time average expectation of the individual utilities and time average expectation of the second auxiliary variables, the second minimum individual time average utility constraint is used for constraining the time average expectation of the individual utilities, the second auxiliary variable constraint is used for constraining the second auxiliary variables, and the third auxiliary variable constraints are used for constraining the time average expectation of the third auxiliary variables and the average time expectation of the individual utilities;

the first selection model includes a first objective function and a first constraint, where the first objective function is a proportional fairness function with time average of the individual utilities as independent variables, and the first constraint includes at least a first coarse correlated equilibrium constraint, a first minimum individual time average utility constraint and a first action probability constraint, where the first coarse correlated equilibrium constraint is used for constraining the time average of the individual utilities and first auxiliary variables, the first minimum individual time average utility constraint is used for constraining the time average of the individual utilities, and the first action probability constraint is used for constraining the action probability under the condition of an random event vector;

the time average of the individual utilities is obtained according to the individual utilities, a random event probability and the action probability under the condition of the random event vector, where the action probability under the condition of the random event vector is the probability that the users execute the action vector under the condition that the random event vector occurs;

an individual utility of each user is obtained according to the action vector and the random event vector; and

the action vector is generated according to the random event vector, and the random event vector is generated according to a capacity model of the cellular network, a capacity model of the drone-cell networks, accessible network sets of the users and a transmission rate model, where the accessible network sets of the users is generated according to a location model of the drone-cell networks and a location model of the users.

In the network selection method and the apparatus provided by the present disclosure, the network selection method includes acquiring the dynamic network model and the dynamic user model, generating the random event vector according to the acquired dynamic network model and the dynamic user model, constructing the first selection model according to the random event vector and the action vector, obtaining the value of the action probability according to the first selection model, and determining the networks that the users choose to access according to the value of the action probability.

The present disclosure constructs a dynamic network model and a dynamic user model, simulating the high dynamic of the drone-cell-user connection, the fluctuation of the network capacity and the time-varying of user traffic, etc. The present disclosure formulates the network selection problem as a problem of a repeated stochastic game, which simulates the competition and interaction among the users well. The method can maximize the total user utility while ensuring the fairness among the users. It solves the problem that the existing game-theory-based approaches cannot solve the network selection problem for the integrated cellular and drone-cell networks because the integrated cellular and drone-cell networks are highly dynamic, and the network state is hard to predict.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a scenario diagram of a network on which a network selection method according to the present disclosure is based;

FIG. 2 is a flowchart illustrating a network selection method according to an exemplary embodiment of the present disclosure;

FIG. 3 is a schematic diagram of a repeated stochastic game structure obeyed by the network selection method according to the embodiment shown in FIG. 2;

FIG. 4 is a schematic structural diagram of a network selection apparatus for the integrated cellular and drone-cell networks according to an exemplary embodiment of the present disclosure;

FIG. 5 is a flowchart of the method performed by a processor in the network selection apparatus for the integrated cellular and drone-cell networks according to the embodiment shown in FIG. 4 of the present disclosure;

FIG. 6 is a schematic diagram illustrating the change of stability variables with time of the network selection method proposed by the present disclosure when the number of users is N=50 and the number of drones is M_d=6;

FIG. 7 is a schematic diagram illustrating the effect of the number of users N on the running time of the network selection method proposed by the present disclosure when the number of drones is M_d=6;

FIG. 8 is a schematic diagram illustrating the effect of the number of drones M_d on the running time of the network selection method proposed by the present disclosure when the number of users is N=50;

FIG. 9 is a schematic diagram illustrating the effect of the number of users N on the total user utilities obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of drones is M_d=6;

FIG. 10 is a schematic diagram illustrating the effect of the number of drones M_d on the total user utilities obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of users is N=50;

FIG. 11 is a schematic diagram illustrating the effect of the number of users N on the Jain's fairness index obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of drones is M_d=6; and

FIG. 12 is a schematic diagram illustrating the effect of the number of drones M_d on the Jain's fairness index obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of users is N=50.

DETAILED IMPLEMENTATION OF THE EMBODIMENTS

In order to make purpose, technical solutions and advantages of embodiments of the present disclosure clearer, the technical solutions of the embodiments of the present disclosure will be described hereunder clearly and comprehensively with reference to accompanying drawings. Apparently, the described embodiments are only a part of embodiments of the present disclosure, rather than all of them. Based on the embodiments of the present disclosure, all other embodiments obtained by those of ordinary skilled in the art without any creative effort shall fall into the protection scope of the present disclosure.

FIG. 1 is a scenario diagram of a network on which a network selection method according to the present disclosure is based. As shown in FIG. 1, the present disclosure investigates a network selection scenario for the integrated cellular and drone-cell networks. In this scenario, a cellular network 103 is constructed to provide radio access to a collection of users ={1, 2, . . . , N} moving randomly and independently in a geographical area of L×W m² over an infinite sequence of time slots t={0, 1, 2, . . . }. Meanwhile, a set of drone-cells are deployed to alleviate the congestion situation of the cellular network. Meanwhile, the present disclosure assumes that every drone-cell connects directly to a ground station moves independently and is deployed at the same and fixed altitude h. Let ={0, 1, 2, . . . , M} be a network set, where j=1 denotes the cellular network, j∈_d={2, . . . , M} denotes the j-th drone-cell network, and j=0 represents an empty network indicating that a user does not access to any network. At each time slot t, a user can select a network to access from the accessible network set. Meanwhile, the present disclosure does not consider the switching cost when a user changes its network access state.

FIG. 2 is a flowchart illustrating a network selection method according to an exemplary embodiment of the present disclosure. As shown in FIG. 2, the network selection method according to the present disclosure includes:

S101: acquiring a dynamic network model and a dynamic user model.

More specifically, the dynamic network model includes at least a location model of drone-cell networks, a capacity model of a cellular network and a capacity model of the drone-cell networks, and the dynamic user model includes at least a location model and a transmission rate model of users;

For the network coverage characteristic, this embodiment assumes that the cellular network can cover the whole considered area and a drone-cell can only cover a small region. Specifically, every drone-cell is assumed to have the same and limited coverage radius (denoted by R_d). Let r_ij denotes a horizontal distance between the i-th user and the j-th drone-cell. The i-th user can access to the j-th drone-cell, if r_ij≤R_d; otherwise, it cannot.

For the location model of the drone-cell networks, this embodiment introduces a smooth turn mobility model with reflection boundary. In this model, each drone will fly with a smooth and random trajectory without sharp turning. Specifically, a drone is assumed to have a constant forward speed V_d (in m/s) and can change its centripetal acceleration randomly. The duration (in seconds) for the drone to maintain its current centripetal acceleration is subject to an exponential distribution with mean 1/λ_d. Meanwhile, the reciprocal of the turning radius (in m) of the drone-cell is subject to a Gaussian distribution with zero-mean and variance σ_d².

For the capacity model of the networks, this embodiment assumes that the network capacity, denoted by C₁(t), of the cellular network at time slot t is subject to a truncated Gaussian distribution N(μ_b,σ_b²) within the interval C₁(t)∈[μ_b−2σ_b,μ_b+2σ_b], where 2σ_b<μ_b. Meanwhile, this embodiment assumes that for each drone-cell network j∈, its capacity at slot t, denoted by C_j(t), is independent and identically distributed (i.i.d.) and is subject to a truncated Gaussian distribution N(μ_c,σ_c²) within the interval C_j(t)∈[μ_c−2σ_c,μ_c+2σ_c], where 2σ_c<μ_c.

For the location model of the users, this embodiment develops a boundary Gauss-Markov mobility model based on the Gauss-Markov mobility model. Specifically, based on the Gauss-Markov mobility model, this embodiment considers a user that moves within a rectangular area

$\left[-\frac{L}{2},\frac{L}{2}\right]\times \left[-\frac{W}{2},\frac{W}{2}\right]$
and reflects on the boundary. Thus, the location (denoted by l_u(t)=((x_u(t),y_u(t))) and the velocity (denoted by v_u (t)=((v_u^x(t),v_u^y(t))) of a user in the boundary Gauss-Markov mobility model can meet the following updated formulas:

$\begin{array}{cc}{l}_u\left(t+1\right)={\left(-1\right)}^k\odot \left(l_u\left(t\right)+v_u\left(t\right)-k\odot \left(L,W\right)\right)& \left(1\right)\\ v_u\left(t+1\right)={\left(-1\right)}^k\odot {\alpha }_u\odot v_u\left(t\right)+\sqrt{1+{\alpha }_u^2}\odot {\sigma }_u\odot w_u\left(t\right)\mathrm{where}k=\left(k_x,k_y\right)=\left(\lfloor \left(x_u\left(t\right)+v_u^x\left(t\right)+\frac{L}{2}\right)/L\rfloor ,\lfloor \left(y_u\left(t\right)+v_u^y\left(t\right)+\frac{W}{2}\right)/W\rfloor \right),\odot & \left(2\right)\end{array}$
denotes the Hadamard product, └⋅┘ denotes the floor operation, α_u=(α_ux,α_uy) is a two-dimensional (2-D) memory level vector, σ_u=(σ_ux,σ_uy) is a 2-D asymptotic standard deviation vector of the velocity, the 2-D memory level vector and the 2-D asymptotic standard deviation vector of the velocity are constants, w_u (t)=(w_u^x(t),w_u^y(t)) represents a 2-D uncorrelated Gaussian process, and w_u^x(t) and w_u^y(t) are independent and are both zero-mean and unit-variance.

For the transmission rate model of the users, this embodiment assumes that for each user i∈, the required data transmission rate at time slot t (denoted by R_i(t)) is i.i.d. and is subject to a truncated Gaussian distribution N(μ_R(t),σ_R(t)²) within the interval R_i(t)∈[μ_R (t)−2σ_R (t),μ_R(t)+2σ_R(t)], where σ_R (t)=ρ_Rμ_R(t) with ρ_R<½. Furthermore, μ_R(t) is a Markov process that taking a value in a finite set {μ₁, μ₂, . . . , μ_Kμ}, and its one-step transfer probability matrix is defined by this embodiment as P=[p_iμ,jμ]_Kμ×Kμ, where p_iμ,jμ is the probability that the μ_R(t) will transfer from the given current value μ_iμ to the value μ_jμ at next time slot.

S102: generating accessible network sets of users according to a location model of the drone-cell networks and a location model of the users; and generating a random event vector according to a capacity model of the cellular network, a capacity model of the drone-cell networks, the accessible network sets of the users and the transmission rate model.

More specifically, this embodiment assumes that the proposed network selection is a repeated stochastic game problem. FIG. 3 is a schematic diagram of a repeated stochastic game structure obeyed by the network selection method according to the embodiment shown in FIG. 2. As shown in FIG. 3, this structure includes an environment 201, a game manager 202 and players 203. This embodiment regards N users as players 203 and network selection strategies taken by the users as actions. At each time slot t∈{0, 1, 2, . . . ,}, each player 203-i∈ can observe a random event ω_i (t)∈Ω_i from the environment 201 and the game manager 202 can observe the whole random event vector ω(t)=(ω₀(t), ω₁(t), . . . , ω_N(t))∈Ω from the environment 201, where ω₀(t)∈Ω₀ represents the random event known only by the game manager 202 and Ω=Ω₀×Ω₁× . . . ×Ω_N. Specifically, in this embodiment, the random event ω₀(t) known only by the game manager 202 includes the capacity of the cellular network and the capacity of the drone-cell networks, i.e., ω₀ (t)=(C₁(t), C₂(t), . . . , C_M(t)). For all i∈, set ω_i(t)={(t),R_i(t)}, where (t) represents the set of accessible non-empty networks of the player 203-i, i.e., (t)={j∈|r_ij(t)≤R_d}∪{1}.

S103: generating an action vector according to the random event vector.

More specifically, the action vector is used for indicating that the users choose to access to the drone-cell networks and/or the cellular network. After observing the random event vector ω(t) at time slot t, the game manager 202 sends a suggestion s_i(t)∈(ω_i(t)) to each player 203-i, where (ω_i(t))=(t)∪{0} is the finite set of actions available to the player 203-i. For example, s_i(t)=j indicates that the game manager 202 suggests the player 203-i choose the network j. Besides, for convenience of description, this embodiment simplifies (ω_i(t)) as (t).

For each player 203-i∈, it will choose an action α_i(t)∈(t) based on the suggestion s_i(t). For example, α_i(t)=j indicates that the player 203-i chooses to access to the network j. This embodiment lets s(t)=(s₁(t), s₂(t), . . . , s_N(t)) and α(t)=(α₁(t), α₂(t), . . . , α_N(t)) denote the suggestion vector and the action vector, respectively, and defines (t)=₁(t)× . . . ×(t).

S104: obtaining an individual utility of each user according to the action vector and the random event vector.

The random event vector ω(t) and the action vector α(t) at time slot t determine the individual utility u_i(t) of each player 203-i. Formally, the individual utility u_i(t) can take the following formula:
u_i(t)=û_i(α(t),ω(t))  (3)

More explicitly, this embodiment utilizes the following expression to define u_i(t).

Definition 1: for all i∈, the individual utility u_i(t) can be defined as:

$\begin{array}{cc}{u}_i\left(t\right)={\hat{u}}_i\left(\alpha \left(t\right),\omega \left(t\right)\right)=\{\begin{array}{cc}0,& {\alpha }_i\left(t\right)=0\\ R_i\left(t\right)f\left({\eta }_i\left(t\right)\right),& \mathrm{otherwise}\end{array}& \left(4\right)\end{array}$
where ƒ(x) is an effective transmission ratio function defined as:

$\begin{array}{cc}f\left(x\right)=\{\begin{array}{cc}1,& 0\le x\le x_b\\ \frac{1}{{\left(1-x_b\right)}^2}{\left(1-x\right)}^2,& x_b<x\le 1\\ 0,& \mathrm{otherwise}\end{array}& \left(5\right)\end{array}$
where x_b is a constant representing the network busyness ratio threshold. Further,

${\eta }_i\left(t\right)=\frac{\sum _{k\in ℐ}^{}1\left\{{\alpha }_k\left(t\right)={\alpha }_i\left(t\right)\right\}{R}_k\left(t\right)}{C_{{\alpha }_i\left(t\right)}}\ge 0$
with 1{α_k(t)=α_i (t)} denoting a 0-1 indicator function. The indicator function equals one, if α_k(t)=α_i(t); otherwise, it equals zero. C_αi(t) represents the capacity of the network corresponding to α_i(t).

This embodiment assumes that the upper bound of the required data transmission rate R_i(t) for each player 203-i is u_i^max. Then, according to Definition 1, û_i(α(t),ω(t)) satisfies the following condition:
0≤û_i(α(t),ω(t))≤u_i^max  (6)

S105: constructing a first selection model.

More specifically, the probability density function (denoted by π[ω]) of the random event vector ω(t) can be defined as the following formula:
π[ω]Pr[ω(t)=ω], ∀ω∈Ω  (7)

where the notation “” means “defined as equal to”.

This embodiment further defines the action probability Pr[α|ω] as a conditional probability density function based on α∈ and ω∈Ω, where =₁× . . . ×_N and

$i_{}=\bigcup _{{\omega }_i\in {\Omega }_i}{i}_{}\left({\omega }_i\right)=𝒥.$
According to the probability theory, the action probability satisfies the first action probability constraint:

$\begin{array}{cc}\mathrm{Pr}\left[\alpha |\omega \right]\ge 0,\forall \alpha \in ,\omega \in \Omega & \left(8\right)\\ \mathrm{Pr}\left[\alpha |\omega \right]=0,\forall \alpha \notin \left(\omega \right),\omega \in \Omega & \left(9\right)\\ \mathrm{Pr}\left[\alpha |\omega \right]=1,\forall \omega \in \Omega & \left(10\right)\end{array}$

where =₁× . . . ×_N, and _i(ω_i) represents the finite set of actions available to the player 203-i after the random event ω_i is observed.

With the definition of Pr[α|ω], this embodiment defines a variable ū_i denoting the time average of the individual utility u_i(t). According to the law of large numbers, if the action vector α(t) is chosen independently at each time slot t according to the same conditional probability density function Pr[α|ω], it can be guaranteed that for all i∈, ū_i, can be written as the following form with probability 1 (w.p. 1):

$\begin{array}{cc}{u}_i=\sum _{\omega \in \Omega }^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\hat{u}}_i\left(\alpha ,\omega \right)& \left(11\right)\end{array}$

In addition, considering that the game manager 202 has an objective of formulating the Pr[α|ω] to maximize the total user utility while ensuring the fairness among users, this embodiment designs an increasing and concave proportional fairness function ϕ(ū₁, ū₂, . . . , ū_N) for the game manager 202 as the first objective function of the first selection model. Explicitly, this embodiment supposes that the proportional fairness function ϕ(ū₁, ū₂, . . . , ū_N) is a sum of logarithmic functions:
ϕ(ū₁,ū₂, . . . ,ū_N)=log₂(ū_i)  (12)

Each player 203-i, however, is interested in maximizing its own time average utility ū_i. Thus, players 203 may choose whether to accept the suggestions provided by the game manager 202. For each player 203-i, there are two types of selections as presented below.

Participate:

if a player 203-i always chooses to accept the suggestion s_i(t) at each time slot t∈{0, 1, 2, . . . ,}, it is called as participate. That is, α_i (t)=s_i(t) for all t∈{0, 1, 2, . . . }.

Non-Participate:

if a player 203-i chooses the action α_i(t) according to its observed random event ω_i(t) at each time slot t∈{0, 1, 2, . . . ,}, it is called as non-participate.

This embodiment assumes that non-participating players 203 will not receive the suggestions s_i(t).

Definition 2: in order to incentivize all players 203 to participate, the Pr[α|ω] formulated by the game manager 202 needs to be a coarse correlated equilibrium (CCE), which is defined as follows:

For a stochastic game, Pr[α|ω] is a CCE if there are first auxiliary variables φ_i(υ_i)∈[0, u_i^max] for all i∈ meeting the following conditions:

$\begin{array}{cc}\sum _{\omega \in \Omega }^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\hat{u}}_i\left(\alpha ,\omega \right)\ge \sum _{\omega \in \Omega }^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\phi }_i\left({\omega }_i\right)& \left(13\right)\\ \sum _{\omega \in \Omega |{\omega }_i={\upsilon }_i}^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\phi }_i\left({\upsilon }_i\right)\ge \sum _{\omega \in \Omega |{\omega }_i={\upsilon }_i}^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\hat{u}}_i\left(\left({\beta }_i,{\alpha }_{\stackrel{\_}{i}}\right),\omega \right)\forall {\upsilon }_i\in {\Omega }_i,{\beta }_i\in i_{}& \left(14\right)\end{array}$

where, α_ī=α\{α_i} represents all entries in the action vector α except for the α_i, β_i∈ is a preset specific action of a non-participating player 203-i, and υ_i∈Ω_i is a preset specific event of a non-participating player 203-i. Intuitively, φ_i(υ_i) represents the largest conditional expected utility earned by the non-participating player 203-i when ω_i=υ_i is observed.

According to Definition 2, the total number of the CCE constraints (13) and (14) is N+|Ω_i∥|, which is a linear function over the size of sets Ω_i and , where |⋅| represents the number of elements in a set. The value of |Ω_i∥|, however, is too large in the system model of this embodiment; thus, the computation of a CCE is complex. This embodiment next describes how to reduce the value of |Ω_i∥|.

First, this embodiment simplifies the value space Ω_i of the preset event υ_i. From the description of the network model in this embodiment, all drone-cells are homogenous. Therefore, for the preset event υ_i, this embodiment only considers the number of the accessible drone-cell networks of a user i∈, rather than the difference in their indexes. Furthermore, the probability that a user i is covered by more than two drone-cell networks simultaneously is small. Thus, the set of (t) can be simplified as {0, 1, 2+}, where, “0”, “1”, and “2+” indicate that the number of the drone-cell networks covering a user i is zero, one and not less than two, respectively. Besides, for preset event υ_i, this embodiment divides the interval [0, u_i^max] into K_R segments uniformly. If

$R_i\left(t\right)\in \left[\frac{\left(i_k-1\right)u_i^{\mathrm{ma}x}}{K_R},\frac{i_k{u}_i^{\mathrm{ma}x}}{K_R}\right),$
where i_k=1, 2, . . . , K_R, then R_i(t) belongs to the i_k-th segment. To sum up, this embodiment can simplify the value space Ω_i of the preset event υ_i as Ω_i^s={0, 1, 2+}×{1, 2, . . . , K_R}. Note that the value space of ω_i is still Ω_i. For all ω_i∈Ω_i and υ_i∈Ω_i^s, ω_i=υ_i in the constraint (14) indicates that ω_i is the original form of υ_i, and in the constraint (13), φ_i(ω_i)=φ_i(υ_i|υ_i=ω_i) with υ_i=ω_i indicating that υ_i is the simplified form of ω_i.

Second, this embodiment simplifies the value space of the preset action β_i. Owing to the homogeneity of drone-cells, this embodiment does not identify the index of the drone-cell network that a user i chooses to access for the preset action β_i. Meanwhile, since the individual utility u_i(t) of a user i at time slot t equals zero when the user i accesses to the empty network according to Definition 1, this embodiment does not consider the preset action of accessing to the empty network, i.e., β_i≠0. Therefore, this embodiment simplifies the value space of the preset action β_i as ={cellular, drone-cell}, where “cellular” and “drone-cell” indicate that the user i chooses to access to a cellular network and a drone-cell network, respectively. Besides, when choosing to access to a drone-cell network, the user i accesses randomly to an accessible drone-cell network with equal probability. Note that the value space of α_i is still .

Therefore, the value of |Ω_i∥| is reduced to |Ω_i^s∥|=(|{0,1,2+}|×K_R)×||=6K_R. Meanwhile, the preset action β_i=drone-cell is infeasible when there is no accessible drone-cell networks for a user i. Thus, this embodiment can ignore the following preset event-preset action pairs: {(0, i_k;drone-cell), ∀i_k∈{1, 2, . . . , K_R} }. In this way, the value of |Ω_i^s∥| is further reduced to 6K_R−K_R=5K_R. Finally, the total number of the CCE constraints (13) and (14) is reduced to N+|Ω_i^s∥|=(5K_R+1)N.

In actual scenarios, some users have a requirement of minimum time average utilities (MTAU), and this embodiment denotes the set of these users as S_u. Therefore, the game manager 202 must guarantee the utilities of these users meet the following MTAU constraints:
ū_i≥u_i^c, ∀i∈S_u  (15)

Based on the above analysis, the constructed first selection model includes: a first objective function and a first constraint. The first constraint includes: a first coarse correlated equilibrium constraint, a first minimum individual time average utility constraint and a first action probability constraint. The constructed first selection model is as follows:

$\begin{array}{cc}\underset{\mathrm{Pr}\left[\alpha |\omega \right]{\phi }_i\left({\upsilon }_i\right)}{\mathrm{Maximize}}\varphi \left({\stackrel{\_}{u}}_1,\dots ,{\stackrel{\_}{u}}_N\right)\mathrm{subject}\mathrm{to}\text{:}{\stackrel{\_}{u}}_i=\sum _{\omega \in \Omega }^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\hat{u}}_i\left(\alpha ,\omega \right),\forall i\in ℐ\sum _{\omega \in \Omega }^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\hat{u}}_i\left(\alpha ,\omega \right)\ge \sum _{\omega \in \Omega }^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\phi }_i\left({\omega }_i\right),\forall i\in ℐ\sum _{\omega \in \Omega |{\omega }_i={\upsilon }_i}^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\phi }_i\left({\upsilon }_i\right)\ge \sum _{\omega \in \Omega |{\omega }_i={\upsilon }_i}^{}\pi \left[\omega \right]\mathrm{Pr}\left[\alpha |\omega \right]{\hat{u}}_i\left(\left({\beta }_i,{\alpha }_{\stackrel{\_}{i}}\right),\omega \right)\forall i\in ℐ,{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in i_s^{}{\stackrel{\_}{u}}_i\ge u_i^c,\forall i\in {𝒮}_u\mathrm{Pr}\left[\alpha |\omega \right]\ge 0,\forall \alpha \in ,\omega \in \Omega \mathrm{Pr}\left[\alpha |\omega \right]=0,\forall \alpha \notin \left(\omega \right),\omega \in \Omega \mathrm{Pr}\left[\alpha |\omega \right]=1,\forall \omega \in \Omega & \left(16\right)\end{array}$

S106: obtaining a value of an action probability according the first selection model, and determining networks that the users choose to access according to the value of the action probability.

The objective of the game manager 202 is to solve the first selection model to obtain the action probability Pr[α|ω], and to choose suggestion vector s(t)=α(t) according to Pr[α|ω] to determine the networks that the users choose to access according to the suggestion vector.

The network selection method provided by this embodiment includes: acquiring the dynamic network model and the dynamic user model, generating the random event vector according to the dynamic network model and the dynamic user model, constructing the first selection model according to the random event vector and the action vector, obtaining the value of the action probability according to the first selection model to determine the networks that the users choose to access according to the value of the action probability. This embodiment solves the problem that the existing game-theory-based approaches cannot solve the network selection problem for the integrated cellular and drone-cell networks because the integrated cellular and drone-cell networks are highly dynamic, and the network state is hard to predict.

Although the above problem (16) is a convex optimization problem, it is highly challenging to solve it owing to the following two reasons: 1) π[ω] is essential to solve this problem (16); however, it may be impossible to obtain π[ω] because π[ω] is influenced by various factors such as network capacity, the mobility of drones and users, and user traffic. 2) The size of the variable Pr[α|ω] is

$\prod _{i\in ℐ}^{}i_{}\prod _{i\in ℐ\bigcup \left\{0\right\}}^{}{\Omega }_i,$
which increases exponentially with the increasing number of users. To solve these issues, this embodiment converts this challenging problem into a new problem without knowing π[ω], and the size of the new problem is greatly reduced.

This embodiment illustrates a network selection method according to another exemplary embodiment, which differs from the embodiment illustrated in FIG. 2 in that, after S104, it includes:

S1051: constructing a second selection model.

More specifically, for a real-valued stochastic process u(t) over time slots t∈{0, 1, 2, . . . }, this embodiment defines its time average expectation over the first t time slots as:

$\begin{array}{cc}\stackrel{\_}{u}\left(t\right)=\frac{1}{t}\sum _{\tau =0}^{t-1}E\left[u\left(\tau \right)\right]& \left(17\right)\end{array}$

For all i∈, υ_i∈Ω_i^s and β_i∈, this embodiment defines:
u_i,υi^(βi)(t)û_i((β_i,α_ī(t)),ω(t))1{ω_i(t)=υ_i}  (18)

By the stochastic process theory, this embodiment converts the first selection model (16) equivalently into the second selection model, where the second selection model includes a second objective function and a second constraint, where the second objective function is a proportional fairness function with the time average expectation of the individual utilities as independent variables, and the second constraint includes at least second coarse correlated equilibrium constraints, a second minimum individual time average utility constraint and a second auxiliary variable constraint, where the second coarse correlated equilibrium constraints are used for constraining the time average expectation of the individual utilities and the time average expectation of second auxiliary variables, the second minimum individual time average utility constraint is used for constraining the time average expectation of the individual utilities, and the second auxiliary variable constraint is used for constraining the second auxiliary variables. At each time slot t∈{0, 1, 2, . . . }, the game manager 202 observes the random event vector ω(t)∈Ω, and solves an action vector α(t)∈(t) and variables φ_i,υi(t), where φ_i,υi(t) represent the second auxiliary variable at time slot t.

$\begin{array}{cc}\underset{\alpha \left(t\right),{\phi }_{i,{\upsilon }_i}}{\mathrm{Maximize}}\underset{t->\infty }{\lim \inf }\varphi \left(u_1\left(t\right),\dots ,{\stackrel{\_}{u}}_N\left(t\right)\right)\mathrm{subject}\mathrm{to}\text{:}\underset{t->\infty }{\lim \inf }\left[{\stackrel{\_}{u}}_i\left(t\right)-\sum _{{\upsilon }_i\in {\Omega }_i^s}^{}{\stackrel{\_}{\phi }}_{i,{\upsilon }_i}\left(t\right)\right]\ge 0,\forall i\in ℐ\underset{t->\infty }{\lim \inf }\left[{\stackrel{\_}{\phi }}_{i,{\upsilon }_i}\left(t\right)-{\stackrel{\_}{u}}_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]\ge 0,\forall i\in ℐ,{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in i_s^{}0\le {\phi }_{i,{\upsilon }_i}\left(t\right)\le u_i^{\mathrm{ma}x}1\left\{{\omega }_1\left(t\right)={\upsilon }_i\right\},\forall t\in \{0,1,2,\dots \},i\in ℐ,{\upsilon }_i\in {\Omega }_i^s\underset{t->\infty }{\lim \inf }\left[{\stackrel{\_}{u}}_i\left(t\right)-u_i^c\left(t\right)\right]\ge 0,\forall i\in {𝒮}_u\alpha \left(t\right)\in \left(t\right),\forall t\in \{0,1,2,\dots \}& \left(19\right)\end{array}$

S106: obtaining a value of the action vector according to the second selection model, and determining the networks that the users choose to access according to the value of the action vector.

In the network selection method provided in this embodiment, the constructed second selection model is based on the time average expectation of the individual utilities, the action vector can be obtained when the probability of the random event vector is unknown, and the networks that the users choose to access are determined according to the value of the action vector.

This embodiment illustrates a network selection method according to another exemplary embodiment, which differs from the previous embodiment in that, after S1051, it further includes:

S1052: constructing a third selection model.

More specifically, the objective of the above problem (19) is to maximize a nonlinear function of time average. In order to convert it equivalently into a maximization of the time average of a nonlinear function, this embodiment introduces a third auxiliary vector γ(t)=(γ₁(t), . . . , γ_N(t)) with 0≤γ_i(t)≤u_i^max for all i∈, and defines g(t)=ϕ(γ₁(t), . . . , γ_N(t). Then, the Jensen's inequality indicates that
g(t)≤ϕ(γ₁(t), . . . ,γ_N(t))  (20)

This embodiment considers converting the second selection model (19) into a third selection model (21.1) by leveraging the Jensen's inequality. Where the third selection model includes a third objective function and a third constraint, and the third objective function (21.1) is time average expectation of a proportional fairness function with the third auxiliary variables as independent variables, and the third constraint includes at least the second coarse correlated equilibrium constraints (21.4) and (21.5), the second minimum individual time average utility constraint (21.7), the second auxiliary variable constraint (21.6) and third auxiliary variable constraints (21.2) and (21.3). At each time slot t∈{0, 1, 2, . . . }, the game manager 202 observes the random event vector ω(t)∈Ω and chooses an action vector α(t)∈(t), variables φ_i,υi(t) and an auxiliary vector γ(t) to solve:

$\begin{array}{cc}\underset{\alpha \left(t\right),{\phi }_{i,{\upsilon }_i}\left(t\right),\gamma \left(t\right)}{\mathrm{Maximize}}\underset{t->\infty }{\lim \inf }\stackrel{\_}{g}\left(t\right)\mathrm{subject}\mathrm{to}\text{:}& \left(21.1\right)\\ \underset{t->\infty }{\lim }{\stackrel{\_}{\gamma }}_i\left(t\right)-{\stackrel{\_}{u}}_i\left(t\right)=0,\forall i\in ℐ& \left(21.2\right)\\ 0\le {\gamma }_i\left(t\right)\le u_i^{\mathrm{ma}x},\forall t\in \{0,1,2,\dots \},i\in ℐ& \left(21.3\right)\\ \underset{t->\infty }{\lim \inf }\left[{\stackrel{\_}{u}}_i\left(t\right)-\sum _{{\upsilon }_i\in {\Omega }_i^s}^{}{\stackrel{\_}{\phi }}_{i,{\upsilon }_i}\left(t\right)\right]\ge 0,\forall i\in ℐ& \left(21.4\right)\\ \underset{t->\infty }{\lim \inf }\left[{\stackrel{\_}{\phi }}_{i,{\upsilon }_i}\left(t\right)-{\stackrel{\_}{u}}_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]\ge 0,\forall i\in ℐ,{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in i_s^{}& \left(21.5\right)\\ 0\le {\phi }_{i,{\upsilon }_i}\left(t\right)\le u_i^{\mathrm{ma}x}1\left\{{\omega }_1\left(t\right)={\upsilon }_i\right\},\forall t\in \{0,1,2,\dots \},i\in ℐ,{\upsilon }_i\in {\Omega }_i^s& \left(21.6\right)\\ \underset{t->\infty }{\lim \inf }\left[{\stackrel{\_}{u}}_i\left(t\right)-u_i^c\right]\ge 0,\forall i\in {𝒮}_u& \left(21.7\right)\\ \alpha \left(t\right)\in \left(t\right),\forall t\in \{0,1,2,\dots \}& \left(21.8\right)\end{array}$

where, the second coarse correlated equilibrium constraints (21.4) and (21.5) are used to constrain the time average expectation of the individual utilities and the time average expectation of the second auxiliary variables, the second minimum individual time average utility constraint (21.7) is used to constrain the time average expectation of the individual utilities, the second auxiliary variable constraint (21.6) is used to constrain the second auxiliary variables, and the third auxiliary variable constraints (21.2) and (21.3) are used to constrain the time average expectation of the third auxiliary variables and the time average expectation of the individual utilities.

S106: obtaining the value of the action vector according to the third selection model, and determining the networks that the users choose to access according to the value of the action vector.

In the network selection method provided in this embodiment, the time average expectation of the proportional fairness function with the third auxiliary variables as independent variables is taken as the third objective function, so that the objective function can be simplified to obtain the action vector according to the third selection model.

This embodiment illustrates a network selection method according to another exemplary embodiment, which differs from the previous embodiment in that, after S1052, it includes:

S1053: converting the third selection model into a fourth selection model by leveraging a drift-plus-penalty technique.

Considering the principle of the drift-plus-penalty technique, as for the constraint (21.1), this embodiment defines the first term Q_i(t) of the first virtual queue for all i∈ as the following form:

$\begin{array}{cc}{Q}_i\left(t+1\right)=Q_i\left(t\right)+\sum _{{\upsilon }_i\in {\Omega }_i^s}^{}{\phi }_{i,{\upsilon }_i}\left(t\right)-u_i\left(t\right)& \left(22\right)\end{array}$

The constraint (21.4) can be satisfied, if the following mean-rate stability condition is held:

$\begin{array}{cc}\underset{t\to \infty }{\lim }\frac{E\left[{\left[Q_i\left(t\right)\right]}^{+}\right]}{t}=0& \left(23\right)\end{array}$

where, the nonnegative operation [x]⁺=max{x,0}.

Likewise, to enforce the constraints (21.1), (21.1) and (21.1), this embodiment defines other three types of virtual queues, respectively. The second item D_i,υi^(βi)(t) of the first virtual queue is defined for all i∈, υ_i∈Ω_i^s, and β_i∈:
D_i,υi^(βi)(t+1)=D_i,υi^(βi)(t)+u_i,υi^(βi)(t)−φ_i,υi(t)  (24)

The second virtual queue Z_i(t) is defined for all i∈:
Z_i(t+1)=Z_i(t)+γ_i(t)−u_i(t)  (25)

The third virtual queue H_i(t) is defined for all i∈S_u:
H_i(t+1)=H_i+u_i^c−u_i(t)  (26)

And the constraints (21.1), (21.1) and (21.1) can be satisfied, if the following mean-rate stability conditions are held:

$\begin{array}{cc}\underset{t\to \infty }{\lim }\frac{E\left[{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}\right]}{t}=0& \left(27\right)\\ \underset{t\to \infty }{\lim }\frac{E\left[Z_i\left(t\right)\right]}{t}=0& \left(28\right)\\ \underset{t\to \infty }{\lim }\frac{E\left[{\left[H_i\left(t\right)\right]}^{+}\right]}{t}=0& \left(29\right)\end{array}$

For simplicity, this embodiment assumes that all virtual queues are initialized to zero.

According the formulas (22), (24), (25) and (26), the first virtual value in the first virtual queue at the current time slot is generated according to the violation of the second coarse correlated equilibrium constraints at the previous time slot and the first virtual value in the first virtual queue at the previous time slot. The second virtual value in the second virtual queue at the current time slot is generated according to the violation of the third auxiliary variable constraints at the previous time slot and the second virtual value in the second virtual queue at the previous time slot. The third virtual value in the third virtual queue at the current time slot is generated according to the violation of the second minimum individual time average utility constraint at the previous time slot and the third virtual value in the third virtual queue at the previous time slot.

This embodiment defines a function L(t) as a sum of squares of all four types of queues [Q_i(t)]⁺, [D_i,υi^(βi)(t)]⁺, Z_i(t) and [H_i(t)]⁺ (divided by 2 for convenience) at time slot t, which is called a Lyapunov function as the total violation:

$\begin{array}{cc}L\left(t\right)\stackrel{△}{=}\frac{1}{2}\sum _{i\in ℐ}^{}{\left({\left[Q_i\left(t\right)\right]}^{+}\right)}^2+\frac{1}{2}\sum _{i\in ℐ}^{}\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }^{}{\left({\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}\right)}^2+\frac{1}{2}\sum _{i\in ℐ}^{}{Z_i\left(t\right)}^2+\frac{1}{2}\sum _{i\in ℐ}^{}{\left({\left[H_i\left(t\right)\right]}^{+}\right)}^2& \left(30\right)\end{array}$

where, H_i(t)=0 for all i∉S_u.

Besides, this embodiment defines a drift-plus-penalty expression as Δ(t)−Vg(t), where Δ(t)=L(t+1)−L(t) represents a Lyapunov drift, that is, the drift value of the total violation, −g(t) is a “penalty”, g(t) is the proportional fairness function with the third auxiliary variables as independent variables, and V is a non-negative penalty coefficient that affects a trade-off between the constraint violation and the optimality. The drift-plus-penalty expression satisfies the following condition: minimizing the constraint violation and maximizing the objective. Therefore, the fourth selection model is constructed as follows:

$\begin{array}{cc}\Delta \left(t\right)-Vg\left(t\right)\le B+\sum _{l\in ℐ}^{}{\left[H_i\left(t\right)\right]}^{+}{u}_l^c-V\varphi \left({\gamma }_1\left(t\right),\dots ,{\gamma }_N\left(t\right)\right)+\sum _{i\in ℐ}^{}{Z}_i\left(t\right){\gamma }_i\left(t\right)+\sum _{i\in ℐ}^{}\left\{{\left[Q_i\left(t\right)\right]}^{+}\sum _{{\upsilon }_i\in {\Omega }_i^s}^{}{\phi }_{i,{\upsilon }_i}\left(t\right)\right\}-\sum _{i\in ℐ}^{}\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }^{}{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}{\phi }_{i,{\upsilon }_i}\left(t\right)+\sum _{i\in ℐ}^{}\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }^{}{\left[D_{l,{\upsilon }_i}^{\left({\beta }_l\right)}\left(t\right)\right]}^{+}{u}_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)-\sum _{i\in ℐ}^{}\left\{{\left[Q_i\left(t\right)\right]}^{+}+Z_i\left(t\right)+{\left[H_i\left(t\right)\right]}^{+}\right\}{u}_i\left(t\right)& \left(31\right)\end{array}$

where a penalty upper bound includes: a constant term, a first penalty upper bound term, a second penalty upper bound term and a third penalty upper bound term. The constant term is

$B\stackrel{△}{=}\sum _{i\in ℐ}^{}{\left(u_i^{\max }\right)}^2+\frac{1}{2}\sum _{i\in S_u}^{}{\left(u_i^{\max }\right)}^2+\frac{1}{2}\sum _{i\in ℐ}^{}{\left(u_i^{\max }\right)}^2,$
the first penalty upper bound term is

$-V\varphi \left({\gamma }_1\left(t\right),\dots ,{\gamma }_N\left(t\right)\right)+\sum _{i\in ℐ}{Z}_i\left(t\right){\gamma }_i\left(t\right),$
the second penalty upper bound term is

$\sum _{i\in ℐ}^{}\left\{{\left[Q_i\left(t\right)\right]}^{+}\sum _{{\upsilon }_i\in {\Omega }_i^s}^{}{\phi }_{i,{\upsilon }_i}\left(t\right)\right\}-\sum _{l\in ℐ}^{}\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }^{}{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}{\phi }_{i,{\upsilon }_i}\left(t\right)$
and the third penalty upper bound term is

$\sum _{i\in ℐ}^{}\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }^{}{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}{u}_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)-\sum _{i\in ℐ}^{}\left\{{\left[Q_i\left(t\right)\right]}^{+}+Z_i\left(t\right)+{\left[H_i\left(t\right)\right]}^{+}\right\}{u}_i\left(t\right).$

S106: obtaining the value of the action vector according the fourth selection model, and determining the networks that the users choose to access according to the value of the action vector.

In the network selection method provided in this embodiment, the fourth selection model is an inequality form, and this model is simple and easy to obtain the action vector according to the fourth selection model.

This embodiment illustrates a network selection method according to another exemplary embodiment, which differs from the previous embodiment in that: S106: obtaining the value of the action vector according the fourth selection model, and determining the networks that the users choose to access according to the value of the action vector, specifically including the following steps:

S1061: at each time slot t, the game manager 202 observes the first term Q_i(t) of the first virtual value in the first virtual queue at the current time slot, the second term D_i,υi^(βi)(t) of the first virtual value in the first virtual queue at the current time slot, the second virtual value Z_i(t) in the second virtual queue at the current time slot, the third virtual value H_i(t) in the third virtual queue at the current time slot and the random event vector ω(t)∈Ω.

This embodiment solves the problem (21.1) by minimizing the upper bound of Δ(t)−Vg(t) at each time slot t greedily. Meanwhile, the upper bound of Δ(t)−Vg(t) can be decomposed into four independent terms. At each time slot t, the first term is a constant, the second term is a function of the third auxiliary vector γ(t), the third term is a function of the second auxiliary variables φ_i,υi(t), and the forth term is a function of the individual utilities u_i(t) and u_i,υi^(βi)(t).

S1062: choosing the value of the third virtual variables γ_i(t) for all i∈ according to the second virtual value at the current time slot and the first penalty upper bound term to solve:

$\begin{array}{cc}\mathrm{Minimize}-V\varphi \left({\gamma }_1\left(t\right),\dots ,{\gamma }_N\left(t\right)\right)+\sum _{i\in ℐ}^{}{Z}_i\left(t\right){\gamma }_i\left(t\right)\mathrm{subject}\mathrm{to}\text{:}0\le {\gamma }_i\left(t\right)\le u_i^{\max },\forall i\in ℐ& \left(32\right)\end{array}$

The closed-form solution of the problem (32) can take the following form for all i∈:

$\begin{array}{cc}{\gamma }_l\left(t\right)=\{\begin{array}{cc}{u}_i^{\max },& Z_i\left(t\right)\le 0\\ \min \left\{\frac{V}{Z_i\left(t\right)\ln 2},u_i^{\max }\right\},& Z_i\left(t\right)>0\end{array}& \left(33\right)\end{array}$

S1063: choosing the value of the second virtual variables φ_i,υi(t) for all i∈and υ_i ∈Ω_i^s according to the random event vector, the first virtual value at the current time slot and the second penalty upper bound term to solve:

$\begin{array}{cc}\mathrm{Minimize}\sum _{i\in ℐ}^{}\left\{{\left[Q_i\left(t\right)\right]}^{+}\sum _{{\upsilon }_i\in {\Omega }_i^s}^{}{\phi }_{i,{\upsilon }_i}\left(t\right)\right\}-\sum _{l\in ℐ}^{}\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }^{}{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}{\phi }_{i,{\upsilon }_i}\left(t\right)\mathrm{subject}\mathrm{to}\text{:}0\le {\phi }_{i,{\upsilon }_i}\left(t\right)\le u_i^{\max }1\left\{{\omega }_i\left(t\right)={\upsilon }_i\right\}\forall i\in ℐ,{\upsilon }_i\in {\Omega }_i^s& \left(34\right)\end{array}$

The closed-form solution of the problem (34) can take the following form for all i∈and υ_i ∈Ω_i^s:

$\begin{array}{cc}{\phi }_{i,{\upsilon }_i}\left(t\right)=\{\begin{array}{cc}{u}_i^{\max },& {\omega }_i\left(t\right)={\upsilon }_i,{\left[Q_i\left(t\right)\right]}^{+}<\sum _{{\beta }_i\in }^{}{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}\\ 0,& \mathrm{otherwise}\end{array}& \left(35\right)\end{array}$

S1064: choosing the value of the action vector α(t) according to the random event vector, the first virtual value at the current time slot, the second virtual value at the current time slot, the third virtual value at the current time slot and the third penalty upper bound term to solve:

$\begin{array}{cc}\mathrm{Maximize}-\sum _{l\in ℐ}^{}\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }^{}{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}{u}_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)+\sum _{i\in ℐ}^{}\left\{{\left[Q_i\left(t\right)\right]}^{+}+Z_i\left(t\right)+{\left[H_i\left(t\right)\right]}^{+}\right\}{u}_i\left(t\right)\mathrm{subject}\mathrm{to}\text{:}\alpha \left(t\right)\in \left(t\right)& \left(36\right)\end{array}$

S1065: sending α_i(t) to each player 203-i, so that the player 203-i determines the network to choose to access according to the action α_i(t).

The individual utilities u_i(t) and u_i,υi^(βi)(t) are computed according to formulas (14) and (18), respectively. The virtual queues Q_i(t), D_i,υi^(βi)(t), Z_i(t) and H_i(t) are updated according to formulas (22), (24), (25) and (26), respectively.

The problem (36) is a non-linear integer programming problem, where u_i(t) and u_i,υi^(βi)(t) are complicate non-linear functions of α(t). The exhaustive algorithm for solving the problem (36) has a complexity of

$O\left(\prod _{i\in ℐ}^{}\left(t\right)\right),$
which increases exponentially with the increasing number of users. Although heuristic algorithms (e.g., genetic algorithm) can be leveraged to mitigate this problem, it may require a long processing time due to its slow convergence rate. To accelerate the optimization process, this embodiment designs a linear approximation mechanism for the problem (36).

According to Definition 1, the network j∈\{0} may be congested when the following condition is held: 1{α_i(t)=j}R_ι(t)>x_bC_j (t). To avoid this situation, the suggested action vector α(t) formulated by the game manager 202 should satisfy the first action vector constraint:

$\begin{array}{cc}\sum _{i\in ℐ}^{}1\left\{{\alpha }_i\left(t\right)=j\right\}{R}_i\left(t\right)\le x_b{C}_j\left(t\right),\forall j\in 𝒥\text{}\left\{0\right\}& \left(37\right)\end{array}$

For each participating player 203-i∈, according to Definition 1, if α_i(t)=0, u_i(t)=0. According to both the constraint (37) and Definition 1, if α_i(t)≠0, u_i(t)=R_i(t). Therefore, the utility function u_i(t) in both cases can be calculated in the following way, thereby forming a mapping table between the individual utilities and the transmission rate of the participating players:
u_i(t)=1{α_i(t)≠0}R_i(t)  (38)

For each non-participating player 203-i∈, and for each υ_i ∈Ω_i^s and β_i∈_i^s, (β_i≠0), this embodiment considers the definition (18) of u_i,υi^(βi)(t). If υ_i≠ω_i(t), u_i,υi^(βi)(t)=0. If υ_i=ω_i(t), u_i,υi^(βi)(t)=û_i((β_i,α_ī(t)), ω(t)). And this embodiment can estimate u_i,υi^(βi)(t) according to the following two different cases, thereby forming a mapping table between the individual utilities and the transmission rate of the non-participating players.

1) If the player 203-i accesses to a network that is just the one suggested by the game manager 202, i.e., β_i=α_i (t), then u_i,υi^(βi)(t)=R_i(t).

2) If the player 203-i accesses to a network that is not the one suggested by the game manager 202, i.e., β_i≠α_i(t), then this embodiment estimates the effective transmission ratio of the network j=β_i at time slot t (denoted by θ_i,βi (t)). Specifically, this embodiment defines the remaining capacity of the network j∈\{0} at time slot t as C_j^r(t)=x_b C_j(t)1{α_i(t)=j}R_i(t), and assumes that C_j^r(t)=C_j^r(t−1). As a result, u_i,υi^(βi)(t) can take the following formula:

$\begin{array}{cc}{u}_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)={\theta }_{i,{\beta }_i}\left(t\right)R_i\left(t\right)& \left(39\right)\end{array}$

where,

${\theta }_{i,{\beta }_i}\left(t\right)=f\left(\frac{{\left[x_b{C}_{{\beta }_i}\left(t\right)-C_{{\beta }_i}^r\left(t-1\right)\right]}^{+}+R_i\left(t\right)}{C_{{\beta }_i}\left(t\right)}\right),$
C_βi(t) represents the capacity of the network corresponding to β_i.

Therefore, u_i,υi^(βi)(t) can be estimated by:

$\begin{array}{cc}{u}_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)=\{\begin{array}{cc}0,& {\upsilon }_i\ne {\omega }_i\left(t\right)\\ R_i\left(t\right),& {\upsilon }_i={\omega }_i\left(t\right),{\beta }_i={\alpha }_i\left(t\right)\\ {\theta }_{i,{\beta }_i}\left(t\right)R_i\left(t\right),& {\upsilon }_i={\omega }_i\left(t\right),{\beta }_i\ne {\alpha }_i\left(t\right)\end{array}& \left(40\right)\end{array}$

Next, this embodiment discusses how to transform the problem (36) into an integer linear programming problem by introducing a set of auxiliary variables a_ij, where {a_ij} is a suggestion matrix. For all i∈ and j∈, this embodiment defines the mapping relationship between the suggestion matrix and the action vector as:

$a_{\mathrm{ij}}=\{\begin{array}{cc}1,& {\alpha }_i\left(t\right)=j\\ 0,& \mathrm{otherwise}\end{array},$
where a_ij=1 indicates that the game manager 202 suggests the player 203-i access to the network j, and a_ij=0 indicates that the game manager 202 suggests the player 203-i not access to the network j. According to the definition of a_ij and the constraint α(t)∈(t), the suggestion matrix constraint can be obtained:

$\begin{array}{cc}\sum _{j\in 𝒥}^{}{a}_{\mathrm{ij}}=1,\forall i\in ℐ& \left(41\right)\\ a_{\mathrm{ij}}\in \left\{0,1\right\},\forall i\in ℐ,j\in 𝒥& \left(42\right)\\ a_{\mathrm{ij}}=0,\forall i\in ℐ,j\notin \left(t\right)& \left(43\right)\end{array}$

α(t) in (37) is substituted with the variables a_ij, and the second action vector constraint can be obtained:

$\begin{array}{cc}\sum _{i\in ℐ}^{}{a}_{\mathrm{ij}}{R}_i\left(t\right)\le x_b{C}_j\left(t\right),\forall j\in 𝒥\text{}\left\{0\right\}& \left(44\right)\end{array}$

α(t) in (37) is substituted with the variables a_ij, and the individual utilities of the participating players can be obtained:

$\begin{array}{cc}{u}_i\left(t\right)=R_i\left(t\right)\sum _{j\in 𝒥\text{}\left\{0\right\}}^{}{a}_{\mathrm{ij}},\forall i\in ℐ& \left(45\right)\end{array}$

Furthermore, this embodiment lets

${\theta }_{i,j}^{\left({\beta }_i\right)}=\{\begin{array}{cc}1,& j={\beta }_i\\ {\theta }_{i,{\beta }_i}\left(t\right),& j\ne {\beta }_i\end{array}$
for all i∈, j∈ and β_i∈. α(t) in (40) is substituted with the variables a_ij, and the individual utilities of the non-participating players can be obtained:
u_i,υi^(βi)(t)û_i((β_i,α_ī(t)),ω(t))1{ω_i(t)=υ_i}, ∀i∈,υ_i∈Ω_i^s,β_i∈_i^s  (46)

According to (41)-(46), this embodiment can transform the problem (36) into the following integer linear programming problem:

$\begin{array}{cc}\underset{\left\{a_{\mathrm{ij}}\right\}}{\mathrm{Maximize}}\sum _{i\in ℐ}^{}\sum _{j\in 𝒥}^{}{c}_{\mathrm{ij}}{a}_{\mathrm{ij}}& \left(47\right)\\ \mathrm{subject}\mathrm{to}\text{:}& \\ \sum _{i\in ℐ}^{}{R}_l\left(t\right)a_{\mathrm{ij}}\le x_b{C}_j\left(t\right),\forall j\in 𝒥\text{}\left\{0\right\}& \\ \sum _{j\in 𝒥}^{}{a}_{\mathrm{ij}}=1,\forall i\in ℐ& \\ a_{\mathrm{ij}}\in \left\{0,1\right\},\forall i\in ℐ,j\in 𝒥& \\ a_{\mathrm{ij}}=0,\forall i\in ℐ,j\notin \left(t\right)& \end{array}$

where, the weights c_ij are defined as:

$\begin{array}{cc}{c}_{\mathrm{ij}}=\{\begin{array}{cc}\left(E_l\left(t\right)-F_{\mathrm{ij}}\left(t\right)\right)R_i\left(t\right),& \forall j\in 𝒥\text{}\left\{0\right\}\\ -F_{\mathrm{ij}}\left(t\right)R_i\left(t\right),& j=0\end{array}& \left(48\right)\\ \mathrm{where}& \\ E_i\left(t\right)={\left[Q_i\left(t\right)\right]}^{+}+Z_i\left(t\right)+{\left[H_i\left(t\right)\right]}^{+}& \\ \mathrm{and}& \\ F_{\mathrm{ij}}\left(t\right)=\sum _{{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in A_i^s}^{}{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}{\theta }_{\mathrm{ij}}^{\left({\beta }_i\right)}1\left\{{\omega }_i\left(t\right)={\upsilon }_i\right\}.& \\ \sum _{i\in ℐ}^{}\sum _{j\in 𝒥}^{}{c}_{\mathrm{ij}}{a}_{\mathrm{ij}}& \end{array}$

is the fourth penalty upper bound term,

$\sum _{i\in ℐ}^{}{R}_i\left(t\right)a_{\mathrm{ij}}\le x_b{C}_j\left(t\right),\forall j\in 𝒥\text{}\left\{0\right\}$
constitutes the second action vector constraint, and

$\{\begin{array}{c}\stackrel{}{{\sum }_{j\in 𝒥}}{a}_{\mathrm{ij}}=1,\forall i\in ℐ\\ a_{\mathrm{ij}}\in \left\{0,1\right\},\forall i\in ℐ,j\in 𝒥\\ a_{\mathrm{ij}}=0,\forall i\in ℐ,j\notin \left(t\right)\end{array}\mathrm{constitutes}\mathrm{the}\mathrm{suggestion}\mathrm{matrix}\mathrm{constraint}.$

Note that at the initial time slot (t=0), all weights c_ij will be zero since all virtual queues are initialized to zero. To handle this problem, this embodiment defines the weights c_ij at time slot t=0 as:

$\begin{array}{cc}{c}_{\mathrm{ij}}=\{\begin{array}{cc}{R}_i\left(0\right),& \forall j\in 𝒥\text{}\left\{0\right\}\\ 0,& j=0\end{array}& \left(49\right)\end{array}$

The problem (47) is an integer linear programming problem with respect to the auxiliary variables a_ij that can be solved by the MOSEK Optimization Tools. Furthermore, in the MOSEK, the branch-and-bound scheme is leveraged to relax the integer variables; thus, the integer linear optimization problem is relaxed to a solvable linear optimization problem.

This embodiment uses the Lyapunov optimization method and the linear approximation mechanism to transform the constructed problem without knowing the state probability π[ω] of the networks and the users, thereby reducing the computational complexity of the problem greatly.

Based on the main framework of solving the problem (21), and combined with the solutions of (32), (34) and (36), this embodiment proposes an network selection apparatus for the integrated cellular and drone-cell networks based on the efficient and fair network selection (EFNS) method shown in FIG. 4.

FIG. 4 is a schematic structural diagram of a network selection apparatus for the integrated cellular and drone-cell networks according to an exemplary embodiment of the present disclosure. As shown in FIG. 4, a network selection apparatus 300 includes a transceiver 311 and a memory 312. The transceiver 311 is used for collecting the drone-cell network capacity information C_j(t), j∈_d, the cellular network capacity information C₁(t), the users' accessible network set information (t), i∈, and data transmission rate information R_i(t), i∈, to form the random event vector ω(t). Meanwhile, the transceiver 311 is also responsible for transmitting the suggested action information formulated by the game manager 202 to each user device 330. The memory 312 can be any type of computer readable medium, and it is responsible for storing information such as parameters, state data, action data, and virtual queue data. The network selection apparatus 300 further includes a processor 313. The processor 313 can be any type of central processing unit, and it is responsible for processing data in the EFNS method. Specifically, the processor 313 obtains the action vector according to the drone-cell network capacity information, the cellular network capacity information, the users' accessible network set information, and data transmission rate information and the fourth selection model. Each user device 330 includes a network access system 331 for controlling the user device 330 to select a network to access according to the received action. The fourth selection model has been described in detail in the foregoing embodiment, and details are not described herein again.

Referring further to FIG. 4, the network selection apparatus provided by this embodiment also includes a human-computer interaction module 314, which includes a display and an operator input interface. The display can display the results to a computer operator 340. The operator input interface can obtain information inputted by the computer operator 340 from one or more input devices such as a keyboard and a mouse.

Referring further to FIG. 4, the network selection apparatus provided by this embodiment further includes networks 320. FIG. 5 is a flowchart of the method performed by a processor in the network selection apparatus for the integrated cellular and drone-cell networks according to the embodiment shown in FIG. 4 of the present disclosure. As shown in FIG. 5, the processor 313 performs the following actions:

The state information ω(t) and the virtual queue information Q_i(t), D_i,υi^(βi)(t), Z_i(t), and H_i (t) at the current time slot t are received. And the output information is the suggested action vector α(t) of the game manager 202 at the current time slot t and the virtual queue information Q_i(t+1), D_i,υi^(βi)(t+1), Z_i(t+1), and H_i(t+1) of the next time slot t+1.

S201: obtaining an upper bound u_i^max of data transmission rate, a segment value K_R, a penalty coefficient V, and initializing a first virtual queue to a third virtual queue.

More specifically, these parameters are stored in the memory 312. These parameters can be given default values, and the computer operator 340 can modify these parameters through the human-computer interaction module 314. The virtual queues Q_i(0)=0, D_i,υi^(βi)(0)=0, Z_i(0)=0 and H_i(0)=0 are initialized, and these virtual queues are stored in the memory 312.

Steps 2-7 are repeated at each time slot, where T is a total number of the time slots.

S202: collecting state information of networks and users to form a random event vector.

More specifically, the processor 313 collects the state information ω(t)∈Ω of the networks and the users through the transceiver 311.

Specifically, ω(t) will be stored in the memory 312 temporarily until the end of Step 6.

S203: obtaining third auxiliary variables γ_i(t) according to a second virtual value at a current time slot and a first penalty upper bound term.

More specifically, for each i∈, the processor 313 calculates the third auxiliary variables γ_i(t) according to the formula (33). The third auxiliary variables γ_i(t) will be temporarily stored in the memory 312 until the end of Step 6.

S204: obtaining second auxiliary variables according to the random event vector at the current time slot, a first virtual value at the current time slot and a second penalty upper bound term.

More specifically, for each i∈ and υ_i ∈Ω_i^s, the processor 313 calculates the second auxiliary variables φ_i,υi(t) by the formula (35). The auxiliary variables φ_i,υi(t) will be temporarily stored in the memory 312 until the end of Step 6.

S205: obtaining a suggestion matrix according to a fourth penalty upper bound, a second action vector constraint and a suggestion matrix constraint, and obtaining an action vector according to the suggestion matrix.

More specifically, the processor 313 obtains the suggestion matrix {a_ij} by solving the problem (47), and obtains the action vector α(t) according to the suggestion matrix {a_ij}. Then the transceiver 311 sends the suggested actions α_i(t) to the network access system 331 of each user device 330-i. The suggested action vector α(t) will be temporarily stored in the memory 312 until the end of Step 6.

S206: calculating individual utilities and the first virtual queue to the third virtual queue.

More specifically, the processor 313 calculates u_i(t) and u_i,υi^(βi)(t) using formulas (4) and (18), respectively, calculates Q_i(t+1) D_i,υi^(βi)(t+1), Z_i(t+L) and H_i(t+1) using formulas (22), (24), (25) and (26), and updates the virtual queues in the memory 312.

S207: judging whether a time slot t reaches a preset time slot value, if not, turning to S208, otherwise, stopping the loop.

S208: updating the time slot t, and turning to S201.

The following context presents a simulation leveraging the EFNS method provided by this embodiment to perform the network selection for the integrated cellular and drone-cell networks.

In order to verify the effectiveness of the network selection method, this embodiment designs three benchmark comparison methods: the cellular-only (CO) method, the random access (RA) method and the on-the-spot offloading (OTSO) method. For the CO method, at each time slot, every user always chooses to access to the cellular network. For the RA method, at each time slot, every user always accesses randomly to an accessible non-empty network with equal probability. For the OTSO method, at each time slot, every user checks whether the drone-cell networks can be access, and if so, the user accesses randomly to an accessible drone-cell network with equal probability; otherwise, the user accesses to the cellular network.

The parameter setting in the simulation is summarized as the following: the size of the considered geographic area is 500×500 m², i.e., L=500 m and V=500 m. The repeated stochastic game lasts for 1000 seconds and let the duration of a time slot be one second; thus, the simulation runs for 1000 episodes, i.e., T=1000. In the location model of the drone-cell networks, the initial locations of the drones are distributed in the considered area independently and uniformly, and their initial heading angles are independent and subject to a uniform distribution on [0,2π), and parameters (V_d, λ_d, σ_d²)=(10, 0.1, 0.02). In the network capacity model, the capacity (in Mb/s) of the cellular network is subject to a truncated Gaussian distribution N_tru(200, 20², ±40), and the capacity (in Mb/s) of each drone-cell network is independent and subject to a truncated Gaussian distribution N_tru(30, 3², ±6). Furthermore, the coverage radius of a drone-cell R_d=100 m.

In the location model of the users, the initial locations of the users are distributed in the considered area independently and uniformly, and the initial velocities of the users are independent and subject to 2-D Gaussian distribution N(0, 0; 2², 2², 0), where parameters α_u=(0.73,0.73) and σ_u=(2,2). In the transmission rate model of the users, a parameter ρ_R=0.2 and the process μ_R(t) (in Mb/s) takes a value from the set {μ₁, μ₂, . . . , μ₅ }={2.5, 5, 7.5, 10, 12.5}. Besides, the one-step transition probability matrix P of μ_R(t) is shown in Table 1:

TABLE 1
One-step transition probability matrix P of μR (t)
0.8	0.2	0	0	0
0.2	0.6	0.2	0	0
0	0.2	0.6	0.2	0
0	0	0.2	0.6	0.2
0	0	0	0.8	2

For the function ƒ(x) in Definition 1, this embodiment sets the network busyness ratio threshold x_b=0.9. For the MTAU constraints, this embodiment sets S_u={1, . . . , 10} and sets u_i^c=6 for all i∈S_u. For the network selection method provided by the present disclosure, set u_i^max=20 for all i∈, K_R=5, and the penalty coefficient V=100.

Meanwhile, this embodiment leverages the following four indexes for performance evaluation of the following proposed method, including:

Queue Stability: this embodiment uses the stability variables defined as

$s_Q\left(t\right)=\frac{\underset{i\in ℐ}{\max }{\left[Q_i\left(t\right)\right]}^{+}}{t},s_D\left(t\right)=\frac{\underset{i\in ℐ,{\upsilon }_i\in {\Omega }_i^s,{\beta }_i\in }{\max }{\left[D_{i,{\upsilon }_i}^{\left({\beta }_i\right)}\left(t\right)\right]}^{+}}{t},s_Z\left(t\right)=\frac{\underset{i\in ℐ}{\max }Z_i\left(t\right)}{t},\mathrm{and}{s}_H\left(t\right)=\frac{\underset{i\in s_u}{\max }{\left[H_i\left(t\right)\right]}^{+}}{t}$
over time slots t=1, 2, . . . , T−1 to measure the stability of the queues of the EFNS method.

Running Time: this is the total time of executing the EFNS method for T=1000 episodes.

Total Utility: this is the total utility of all users during the entire simulation process,

$i.e.,\sum _{t=0}^{T-1}\sum _{i\in ℐ}^{}{u}_i\left(t\right).$

Fairness: this embodiment uses the Jain's fairness index, defined as

${\left(\sum _{l\in ℐ}^{}{\stackrel{\_}{u}}_i\right)}^2\text{/}\left(N\sum _{i\in ℐ}^{}{\stackrel{\_}{u}}_l^2\right),$
to measure the fairness of the network resource allocation, where ū_i represents the time average utility of a user i during the entire simulation process, i.e.,

${\stackrel{\_}{u}}_i=\frac{1}{T}{\sum }_{t=0}^{T-1}{u}_i\left(t\right).$

In the simulation, this embodiment tests all comparison methods on one hundred randomly generated data sets. For each comparison method, this embodiment may obtain one hundred results, and the final result is their average.

FIG. 6 illustrates the change of stability variables with time when the network selection method according to the present disclosure is used and when the number of users is N=50 and the number of drones is M_d=6. As shown in FIG. 6:

all stability variables decrease rapidly with the increase of time slot t. After a long period of time, all stability variables tend to be zero. This result indicates that the EFNS method can guarantee that all queues are mean-rate stable, and thus the constraints (21.1), (21.1), (21.1) and (21.1) are satisfied.

FIG. 7 illustrates the effect of the number of users IN on the running time of the network selection method proposed by the present disclosure when the number of drones is M_d=6. FIG. 8 illustrates the effect of the number of drones M_d on the running time of the network selection method proposed by the present disclosure when the number of users is N=50. As shown in FIG. 7 and FIG. 8:

the average running time of the EFNS method increases with the increase of N or M_d. This is because when N or M_d increases, the scale of the problem becomes larger. Meanwhile, the EFNS method can achieve an online network selection.

FIG. 9 illustrates the effect of the number of users N on the total user utilities obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of drones is M_d=6. FIG. 10 illustrates the effect of the number of drones M_d on the total user utilities obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of users is N=50. As shown in FIG. 9 and FIG. 10:

the EFNS method can always achieve the highest total utility compared with the other three methods by avoiding congestion and making full use of network resources.

For the EFNS method, the total utility thereof increases with the increase of N. However, the rate of increase slows down because network capacity limit the increase of the total utility when N is large. For the other three methods, these total utilities thereof begin quickly to decrease with the increase of N. This is because these three methods do not have a mechanism to avoid network congestion and a large number of users may lead to network congestion.

For all methods except for the CO method, their total utilities increase monotonically with the increase of M_d, since users can offload traffic to drone-cell networks.

FIG. 11 illustrates the effect of the number of users N on the Jain's fairness indices obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of drones is M_d=6. FIG. 12 illustrates the effect of the number of drones M_d on the Jain's fairness index obtained by the network selection method proposed by the present disclosure and the comparison methods when the number of users is N=50. As shown in FIG. 11 and FIG. 12:

the EFNS method can achieve a high level of fairness. Specifically, its fairness index is close to 1 because the fair allocation of network resources is considered in the proportional fairness function (12). However, the fairness index of the EFNS method decreases gradually with the increase of N or the decrease of M_d, that is because a great N or a small M_d may activate the MTAU constraints.

Although the RA and OTSO methods do not consider fairness, they can also achieve a high level of fairness. This is because the time average utilities of users are close to each other after a long time due to the homogeneity of users in the model of this embodiment.

The CO method can achieve the highest level of fairness since users always have the same effective transmission ratio at each time slot. This CO method, however, achieves the lowest total utility.

Finally, it should be noted that each of the above embodiments is merely intended to describe, rather than limit, the technical solutions of the present disclosure. Although the present disclosure is described in detail with reference to the foregoing embodiments, persons of ordinary skill in the art should understand that it is possible to make modifications to the technical solutions described in the foregoing embodiments, or make equivalent substitutions of part or all technical features therein. However, these modifications or substitutions do not make the essence of corresponding technical solutions depart from the scope of the technical solutions in the embodiment solutions of the present disclosure.

Claims

1. A network selection method for integrated cellular and drone-cell networks, comprising:

acquiring a dynamic network model and a dynamic user model, wherein the dynamic network model comprises at least a location model of the drone-cell networks, a capacity model of the cellular network and a capacity model of the drone-cell networks, and the dynamic user model comprises at least a location model and a transmission rate model of users;

generating accessible network sets of the users according to the location model of the drone-cell networks and the location model of the users;

generating a random event vector according to the capacity model of the cellular network, the capacity model of the drone-cell networks, the accessible network sets of the users and the transmission rate model, wherein the accessible network sets of the users comprise the drone-cell networks and/or the cellular network;

generating an action vector according to the random event vector, wherein the action vector for indicating that the users choose to access to the drone-cell networks and/or the cellular network;

obtaining an individual utility of each user according to the action vector and the random event vector;

constructing a first selection model, wherein the first selection model comprises a first objective function and a first constraint, wherein the first objective function is a proportional fairness function with the time average of individual utilities of the users as independent variables, and the first constraint comprises at least a first coarse correlated equilibrium constraint, a first minimum individual time average utility constraint and a first action probability constraint, wherein the first coarse correlated equilibrium constraint is used for constraining the time average of the individual utilities and first auxiliary variables, the first minimum individual time average utility constraint is used for constraining the time average of the individual utilities, and the first action probability constraint is used for constraining the action probability under the condition of the random event vector; wherein the time average of the individual utilities is obtained according to the individual utilities, the random event probability and the action probability under the condition of the random event vector, wherein the action probability under the condition of the random event vector is the probability that the users execute the action vector under the condition that the random event vector occurs, and the random event probability is the probability that the random event vector occurs; and

obtaining a value of the action probability according the first selection model, and determining networks that the users choose to access according to the value of the action probability.

2. The method according to claim 1, wherein after the obtaining an individual utility of each user according to the action vector and the random event vector, the method further comprises:

constructing a second selection model, wherein the second selection model comprises a second objective function and a second constraint, wherein the second objective function is a proportional fairness function with time average expectation of the individual utilities as independent variables, and the second constraint comprises at least second coarse correlated equilibrium constraints, a second minimum individual time average utility constraint and a second auxiliary variable constraint, wherein the second coarse correlated equilibrium constraints are used for constraining the time average expectation of the individual utilities and time average expectation of the second auxiliary variables, the second minimum individual time average utility constraint is used for constraining the time average expectation of the individual utilities, and the second auxiliary variable constraint is used for constraining the second auxiliary variables; and

obtaining a value of the action vector according the second selection model, and determining networks that the users choose to access according to the value of the action vector.

3. The method according to claim 2, wherein after the obtaining an individual utility of each user according to the action vector and the random event vector, the method further comprises:

constructing a third selection model according to the second selection model, wherein the third selection model comprises a third objective function and a third constraint, wherein the third objective function is time average expectation of a proportional fair function with third auxiliary variables as independent variables, and the third constraint comprises at least the second coarse correlated equilibrium constraints, the second minimum individual time average utility constraint, the second auxiliary variable constraint, and third auxiliary variable constraints, wherein the second coarse correlated equilibrium constraints are used for constraining the time average expectation of the individual utilities and the time average expectation of the second auxiliary variables, the second minimum individual time average utility constraint is used for constraining the time average expectation of the individual utilities, the second auxiliary variable constraint is used for constraining the second auxiliary variables, and the third auxiliary variable constraints are used for constraining time average expectation of the third auxiliary variables and the time average expectation of the individual utilities;

obtaining a value of the action vector according the third selection model, and determining networks that the users choose to access according to the value of the action vector.

4. The method according to claim 3, wherein after the constructing a third selection model, the method further comprises:

constructing a fourth selection model;

obtaining a value of the action vector according the fourth selection model, and determining networks that the users choose to access according to the value of the action vector;

wherein the fourth selection model is that a difference value between a drift of total violation and an utility is less than or equal to a penalty upper bound;

the drift of the total violation is obtained according to a value of the total violation at a current time slot and a value of the total violation at a next time slot;

the value of the total violation at the current time slot is obtained according to a first virtual value at the current time slot, a second virtual value at the current time slot and a third virtual value at the current time slot;

the first virtual value in a first virtual queue at the current time slot is generated according to violation of the second coarse correlated equilibrium constraints at the previous time slot and the first virtual value in the first virtual queue at the previous time slot;

the second virtual value in a second virtual queue at the current time slot is generated according to violation of third auxiliary variable constraints at the previous time slot and the second virtual value in the second virtual queue at the previous time slot;

the third virtual value in a third virtual queue at the current time slot is generated according to the violation of a second minimum individual time average utility constraint at the previous time slot and the third virtual value in the third virtual queue at the previous time slot, wherein the first virtual value at initial time slot, the second virtual value at the initial time slot, and the third virtual value at the initial time slot are all zero;

the utility comprises: the proportional fair function and a penalty coefficient, with the third auxiliary variables as independent variables; and

the penalty upper bound comprises: a constant term, a first penalty upper bound term, a second penalty upper bound term and a third penalty upper bound term, wherein the first penalty upper bound term comprises the third auxiliary variables and the second virtual value, the second penalty upper bound term comprises the second auxiliary variables and the first virtual value, and the third penalty upper bound comprises the individual utilities, the first virtual value, the second virtual value and the third virtual value.

5. The method according to claim 4, wherein the obtaining a value of the action vector according the fourth selection model, and determining networks that the users choose to access according to the value of the action vector, specifically comprises:

obtaining the first virtual value at the current time slot, the second virtual value at the current time slot, the third virtual value at the current time slot and the random event vector at the current time slot;

obtaining values of the third auxiliary variables according to the second virtual value at the current time slot and the first penalty upper bound term;

obtaining values of the second auxiliary variables according to the random event vector at the current time slot, the first virtual value at the current time slot and the second penalty upper bound term;

obtaining the value of the action vector according to the random event vector at the current time slot, the first virtual value at the current time slot, the second virtual value at the current time slot, the third virtual value at the current time slot and the third penalty upper bound term, and determining the networks that the users choose to access according to the value of the action vector;

the obtaining the value of the action vector according to the random event vector at the current time slot, the first virtual value at the current time slot, the second virtual value at the current time slot, the third virtual value at the current time slot and the third penalty upper bound term, specifically comprises:

constructing a suggestion matrix constraint and a first action vector constraint, wherein a suggestion matrix represents that the users are suggested to access to the drone-cell networks and/or the cellular network, and the suggestion matrix constraint is used for constraining the suggestion matrix;

using a mapping table between the individual utilities and transmission rate and a mapping relationship between the suggestion matrix and the action vector to process the third penalty upper bound term, to generate a fourth penalty upper bound term;

using the mapping relationship between the suggestion matrix and the action vector to process the first action vector constraint, to obtain a second action vector constraint; and

obtaining a value of the suggestion matrix according the fourth penalty upper bound term, the suggestion matrix constraint and the second action vector constraint, and determining the networks that the users choose to access according to the value of the suggestion matrix.

6. The method according to claim 1, wherein the first objective function specifically comprises: ƒ1=ϕ(ū1,..., ui,..., uN), wherein, ƒ1 represents the first objective function, ū1, represents the time average of the individual utility of the i -th user, and ϕ(ū1,..., ui,..., uN)represents a proportional fair function; ∑ ω ∈ Ω ⁢ ∑ α ∈ A ⁢ π ⁡ [ ω ] ⁢ Pr ⁡ [ α | ω ] ⁢ u ^ i ⁡ ( α, ω ) ≥ ∑ ω ∈ Ω ⁢ ∑ α ∈ A ⁢ π ⁡ [ ω ] ⁢ Pr ⁡ [ α | ω ] ⁢ φ i ⁡ ( ω i ), ∀ i ∈ ℐ ⁢ ∑ ω ∈ Ω | ω i = υ i ⁢ ∑ α ∈ A ⁢ π ⁡ [ ω ] ⁢ Pr ⁡ [ α | ω ] ⁢ φ i ⁡ ( υ 1 ) ≥ ∑ ω ∈ Ω | ω i = υ i ⁢ ∑ α ∈ A ⁢ π ⁡ [ ω ] ⁢ Pr ⁡ [ α | ω ] ⁢ u ^ l ⁡ ( ( β i, α i _ ), ω ); ∀ i ∈ ℐ, υ i ∈ Ω i s, β i ∈ A i s Pr ⁡ [ α | ω ] ≥ 0, ∀ α ∈ A, ω ∈ Ω Pr ⁡ [ α | ω ] = 0, ∀ α ∉ A, ( ω ), ω ∈ Ω; ∑ α ∈ A ⁢ Pr ⁡ [ α | ω ] = 1, ∀ ω ∈ Ω

the first coarse correlated equilibrium constraint specifically comprises:

wherein, ω represents the random event vector, α represents the action vector, π[ω] represents a probability of the random event vector, Pr[α|ω] represents an action probability under the condition of the random event vector, û1(α,ω) represents the individual utility of the i-th user, both φi(ωi) and φi) (υi) represent the first auxiliary variable, αī=α\{αi}, αi represents that the i -th user chooses to access to the network j=αi, ωi, represents the i -th element in the random event vector, υi represents a preset event of the i-th user, βi represents a preset action of the i-th user, A represents an available set of the action vector, Ω represents an available set of the random event vector, Ais represents a simplified available set of the preset action of the i-th user, and Ωis represents a simplified available set of the preset event of the i-th user;

the first minimum individual time average utility constraint comprises: ūi≥uic, ∀i∈Su, wherein, uic represents a first minimum individual time average utility, and Su represents a set of the users with minimum individual time average utility requirements; and

the first action probability constraint specifically comprises:

wherein, A(ω) represents the available set of the action vector under the random event vector ω.

7. The method according to claim 3, wherein f 2 = lim ⁢ ⁢ inf t → ∞ ⁢ ϕ ⁡ ( u _ 1 ⁡ ( t ), … ⁢, u _ i ⁡ ( t ), … ⁢, u _ N ⁡ ( t ) ), wherein, ū,(t) represents the time average expectation of the individual utility of the i-th user; lim ⁢ ⁢ inf t → ∞ [ u _ i ⁡ ( t ) - ∑ υ i ∈ Ω i s ⁢ φ _ i, υ i ⁡ ( t ) ] ≥ 0, ∀ i ∈ ℐ lim ⁢ ⁢ inf t → ∞ ⁡ [ φ _ i, υ i ⁡ ( t ) - u _ i, υ i ( β i ) ⁡ ( t ) ] ≥ 0, ∀ i ⁢ ∈ ℐ, υ i ∈ Ω i s, β i ∈ A i s; lim ⁢ ⁢ inf t → ∞ ⁡ [ φ _ i ⁡ ( t ) - u i c ] ≥ 0, ∀ i ⁢ ∈ 𝒮 u; f 3 = lim ⁢ ⁢ inf t → ∞ ⁢ g _ ⁡ ( t ), wherein, g(t)=ϕ(γ1(t),..., γN(t)), 0≤γi(t)≤uimax, and uimax represents an upper bound of data transmission rate required by the i-th user; lim ⁢ ⁢ inf t → ∞ [ u _ i ⁡ ( t ) - ∑ υ i ∈ Ω i s ⁢ φ _ i, υ i ⁡ ( t ) ] ≥ 0, ∀ i ∈ ⁢ ℐ lim ⁢ ⁢ inf t → ∞ ⁡ [ φ _ i, υ i ⁡ ( t ) - u _ i, υ i ( β i ) ⁡ ( t ) ] ≥ 0, ∀ i ⁢ ∈ ℐ, υ i ∈ Ω i s, β i ∈ A i s; lim ⁢ ⁢ inf t → ∞ ⁡ [ u _ i ⁡ ( t ) - u i c ] ≥ 0, ∀ i ⁢ ∈ 𝒮 u; lim ⁢ t → ∞ | γ _ i ⁡ ( t ) - u _ i ⁡ ( t ) | = 0, ∀ i ∈ ℐ 0 ≤ γ i ⁡ ( t ) ≤ u i max, ∀ t ∈ { 0, 1, 2, … ⁢ }, i ∈ ℐ;

the second objective function specifically comprises:

the second coarse correlated equilibrium constraints specifically comprise:

wherein, φi,υi(t) represents the time average expectation of the second auxiliary variables, and ui,υi(βi)(t)□ûi((βi,αī(t)),ω(t))1{ωi(t)=υi}, and 1{⋅} represents an indicator function;

the second minimum individual time average utility constraint specifically comprises:

the second auxiliary variable constraint specifically comprises: 0≤φi,υi(t)≤uimax1{ωi(t)=υi} ∀t∈{0,1,2,... },i∈,υi∈Ωis;

the third objective function specifically comprises:

the second coarse correlated equilibrium constraints specifically comprise:

the second minimum individual time average utility constraint specifically comprises:

the second auxiliary variable constraint specifically comprises: 0≤φi,υi(t)≤uimax1{ωi(t)=υi} ∀t∈{0,1,2,... },i∈,υi∈Ωis;

the third auxiliary variable constraints specifically comprise:

wherein, γi(t) represents the time average expectation of the third auxiliary variables.

8. The method according to claim 4, wherein Δ ⁡ ( t ) - Vg ⁡ ( t ) ≤ B + ∑ i ∈ ℐ ⁢ [ H i ⁡ ( t ) ] + ⁢ u i c - V ⁢ ⁢ ϕ ⁡ ( γ 1 ⁡ ( t ), … ⁢, γ N ⁡ ( t ) ) + ∑ i ∈ ℐ ⁢ Z i ⁡ ( t ) ⁢ γ i ⁡ ( t ) + ∑ i ∈ ℐ ⁢ { [ Q i ⁡ ( t ) ] + ⁢ ∑ υ i ∈ Ω i s ⁢ φ i, υ i ⁡ ( t ) } - ∑ i ∈ ℐ ⁢ ∑ υ i ∈ Ω i s, β i ∈ A i s ⁢ [ D i, υ i ( β i ) ⁡ ( t ) ] + ⁢ φ i, υ i ⁡ ( t ); ⁢ + ∑ i ∈ ℐ ⁢ ∑ υ i ∈ Ω i s, β i ∈ A i s ⁢ [ D i, υ i ( β i ) ⁡ ( t ) ] + ⁢ u i, υ i ( β i ) ⁡ ( t ) - ∑ i ∈ ℐ ⁢ { [ Q i ⁡ ( t ) ] + + Z i ⁡ ( t ) + [ H i ⁡ ( t ) ] + } ⁢ u i ⁡ ( t ) ⁢ wherein, L ⁡ ( t ) ⁢ = △ ⁢ 1 2 ⁢ ∑ i ∈ ℐ ⁢ ( [ Q i ⁡ ( t ) ] + ) 2 + 1 2 ⁢ ∑ i ∈ ℐ ⁢ ∑ υ i ∈ Ω i s, β i ∈ A i s ⁢ ( [ D i, υ i ( β i ) ⁡ ( t ) ] + ) 2 + 1 2 ⁢ ∑ i ∈ ℐ ⁢ Z i ⁡ ( t ) 2 + 1 2 ⁢ ∑ i ∈ ℐ ⁢ ( [ H i ⁡ ( t ) ] + ) 2, Δ(t)=L(t+1)−L(t), [x]+=max{x,0}, Qi(t) represents a value of the first term of the first virtual value at time slot t, Q i ⁡ ( t + 1 ) = Q i ⁡ ( t ) + ∑ υ i ∈ Ω i s ⁢ φ i, υ i ⁡ ( t ) - u i ⁡ ( t ), D i, υ i ( β i ) ⁡ ( t ) represents a value of the second term of the first virtual value at time slot t, Di,υi(βi)(t+1)=Di,υi(βi)(t)+ui,υi(βi)(t)−φi,υi(t), φi,υi(t) represents a value of the second auxiliary variable at time slot t, Zi(t) represents a value of the second virtual value at time slot t, Zi(t+1)=Zi(t)+γi(t)−ui(t), ui(t) represents an individual utility of the i-th user at time slot t, B ⁢ ⁢ • ⁢ ∑ i ∈ ℐ ⁢ ( u i max ) 2 + 1 2 ⁢ ∑ i ∈ δ u ⁢ ( u i max ) 2 + 1 2 ⁢ ∑ i ∈ ℐ ⁢  A i s  ⁢ ( u i max ) 2, |⋅| represents the number of elements in a set, Hi(t) represents a value of the third virtual value at time slot t, and Hi(t+1)=Hi(t)+uic−ui(t).

the fourth selection model specifically comprises:

9. The method according to claim 5, wherein ∑ i ∈ ℐ ⁢ ∑ j ∈ 𝒥 ⁢ c ij ⁢ a ij; ∑ i ∈ ℐ ⁢ R i ⁡ ( t ) ⁢ a ij ≤ x b ⁢ C j ⁡ ( t ), ∀ j ∈ 𝒥 ⁢ \ ⁢ { 0 } ∑ j ∈ 𝒥 ⁢ a ij = 1, ∀ i ∈ ℐ a ij ∈ { 0, 1 }, ∀ i ∈ ℐ, j ∈ 𝒥; a ij = 0, ∀ i ∈ ℐ, j ∉ A i ⁡ ( t ) wherein, c ij = { ( E i ⁡ ( t ) - F ij ⁡ ( t ) ) ⁢ R i ⁡ ( t ), ∀ j ∈ 𝒥 ⁢ \ ⁢ { 0 } - F ij ⁡ ( t ) ⁢ R i ⁡ ( t ), j = 0, E i ⁡ ( t ) = [ Q i ⁡ ( t ) ] + + Z i ⁡ ( t ) + [ H i ⁡ ( t ) ] +, F ij ⁡ ( t ) = ∑ υ i ∈ Ω i s, β i ∈ A i s ⁢ [ D i, υ i ( β i ) ⁡ ( t ) ] + ⁢ θ ij ( β i ) ⁢ 1 ⁢ { ω i ⁡ ( t ) = υ i }, θ i, j ( β i ) = { 1, j = β i θ i, β i ⁡ ( t ), j ≠ β i, represents an element in the suggestion matrix, J represents a set of networks, j represents a serial number of a network, θi,βi(t) is an effective transmission ratio of the i-th user accessing to the network j=βi at time slot t, Ri(t) represents transmission rate of the i-th user at time slot t, Cj(t) represents a capacity of the j-th network at time slot t, and Ai(t) represents an available set of the action of the i-th user at time slot t.

the fourth penalty upper bound term specifically comprises:

the suggestion matrix constraint specifically comprises: